{"id":5198,"date":"2014-11-03T09:00:54","date_gmt":"2014-11-03T17:00:54","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=5198"},"modified":"2020-01-15T10:48:28","modified_gmt":"2020-01-15T18:48:28","slug":"challenging-gmat-problems-with-exponents-and-roots","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/","title":{"rendered":"Challenging GMAT Problems with Exponents and Roots"},"content":{"rendered":"

This post was updated in 2024 for the new GMAT.<\/strong><\/p>\n

Here are twelve challenging problems related to the topic of exponents & roots.\u00a0 Remember, no calculator<\/a>.<\/p>\n

\"cgpwe_img1\"<\/a><\/p>\n

\"cgpwe_img2\"<\/a><\/p>\n

(A) 17<\/p>\n

(B) 19<\/p>\n

(C) 21<\/p>\n

(D) 23<\/p>\n

(E) 27<\/p>\n

\"cgpwe_img3\"<\/a><\/p>\n

4) Rank the following quantities in order, from smallest to biggest.<\/p>\n

\"cgpwe_img4\"<\/a><\/p>\n

(A) I, II, III<\/p>\n

(B) I, III, II<\/p>\n

(C) II, I, III<\/p>\n

(D) III, I, II<\/p>\n

(E) III, II, I<\/p>\n

5) Rank the following quantities in order, from smallest to biggest.<\/p>\n

\"cgpwe_img5\"<\/a><\/p>\n

(A) I, II, III<\/p>\n

(B) I, III, II<\/p>\n

(C) II, I, III<\/p>\n

(D) III, I, II<\/p>\n

(E) III, II, I<\/p>\n

\"cgpwe_img6\"<\/a><\/p>\n

(A) 96<\/p>\n

(B) 120<\/p>\n

(C) 144<\/p>\n

(D) 192<\/p>\n

(E) 288<\/p>\n

\"cgpwe_img7\"<\/a><\/p>\n

(A) 4,000<\/p>\n

(B) 8,000<\/p>\n

(C) 16,000<\/p>\n

(D) 25,000<\/p>\n

(E) 125,000<\/p>\n

 <\/p>\n

8) Rank the following quantities in order, from smallest to biggest.<\/p>\n

\"cgpwe_img8\"<\/a><\/p>\n

(A) I, II, III<\/p>\n

(B) I, III, II<\/p>\n

(C) II, I, III<\/p>\n

(D) II, III, I<\/p>\n

(E) III, II, I<\/p>\n

\"cgpwe_img9\"<\/a><\/p>\n

\"cgpwe_img10\"<\/a><\/p>\n

\"cgpwe_img11\"<\/a><\/p>\n

\"cgpwe_img12\"<\/a><\/p>\n

(A) 1\/5<\/p>\n

(B) 2\/13<\/p>\n

(C) 2\/15<\/p>\n

(D) 5\/3<\/p>\n

(E) 15\/2<\/p>\n

Exponents and roots<\/h2>\n

For a review of some of the basics, see these blogs:<\/p>\n

1) Exponent Properties on the GMAT<\/a><\/p>\n

2) Adding and Subtracting Powers on the GMAT<\/a><\/p>\n

3) Roots<\/a><\/p>\n

4) Dividing by a Square Root<\/a><\/p>\n

5) Practice Problems on Powers and Roots<\/a><\/p>\n

If reading any of those blogs gives you some insight, you might want to give the problems a second look before proceeding the solutions below.<\/p>\n

\"cgpwe_img13\"<\/a><\/p>\n

 <\/p>\n

Practice problems explanations<\/h2>\n

1) For this one, we can eliminate choices (A)<\/strong> & (B)<\/strong> because they make ghastly exponent mistakes.\u00a0 For the other three, we would have to add and subtract the powers<\/a> of each, to see which work, trying each by trial and error.<\/p>\n

Instead, here’s an elegant solution.\u00a0 Notice that the product \"5*2^4. So, this is 80 times a power of 3.\u00a0 Now, notice that 80 = 81 \u2013 1.\u00a0 In other words, we can easily express the factor 80 as the difference of two powers of 3.\u00a0 Thus.<\/p>\n

\"cgpwe_img14\"<\/a><\/p>\n

Answer = (E)<\/strong>.<\/p>\n

2) We have to begin with some clever noticing.\u00a0 Notice, first of all, that 409 = 400 + 9.\u00a0 This suggests a difference of two squares factoring<\/a> pattern.\u00a0 Notice that the numerator can be expressed as 159,919 = 160,000 \u2013 81.\u00a0 That’s a difference of two squares that can be factored!<\/p>\n

\"159,919<\/p>\n

The first parenthesis produces 409.\u00a0 The second parenthesis contains another difference of two squares:<\/p>\n

\"159,919<\/p>\n

\"159,919<\/p>\n

Therefore, if we divide by sides by (409)(17), we get that the fraction equals 23.<\/p>\n

Answer = (D)<\/strong>.<\/p>\n

3) Well, we know that 15 = 3*5, and we have the exponent that gives a 5, but how do we get an exponent that gets a 3?\u00a0 Well, of course [pmath]6^1 = 6[\/pamth], and therefore<\/p>\n

\"cgpwe_img15\"<\/a><\/p>\n

and therefore<\/p>\n

\"cgpwe_img16\"<\/a><\/p>\n

Answer = (E)<\/strong>.<\/p>\n

4) First of all, the fact that 120 is a multiple of 30 draws our attention to that comparison.\u00a0 The ratio 120:30 = 4:1, so we just would have to compare those powers of 2 and 17.<\/p>\n

\"cgpwe_img17\"<\/a><\/p>\n

So, III is bigger than I.\u00a0 Now, notice that the exponents of I and II are also in a convenient ratio — 120:72 = 5:3.\u00a0 We can use those exponents on 2 and 3.<\/p>\n

\"cgpwe_img18\"<\/a><\/p>\n

Therefore, II is smaller than I.\u00a0 From smallest to biggest: II, I, III.<\/p>\n

Answer = (C)<\/strong>.<\/p>\n

5) First of all, clearly<\/p>\n

\"cgpwe_img19\"<\/a><\/p>\n

So, II is bigger than I.\u00a0 Now, what about III?\u00a0 When we take higher order roots, the values move closer to one.\u00a0 If the number starts larger than one, then higher and higher roots make it smaller, closer to one.\u00a0 If the number starts between 0 and 1, then higher and higher roots make it larger, closer to one.\u00a0 Therefore, III is larger than II.\u00a0 From smallest to biggest, I, II, III.<\/p>\n

Answer = (A)<\/strong>.<\/p>\n

6) First of all, the mistake: we CANNOT add through<\/p>\n

\"cgpwe_img20\"<\/a><\/p>\n

That incorrect thinking would lead to the trap answer of (B)<\/strong>.\u00a0 Instead, we have to simplify each square root on the left.<\/p>\n

\"cgpwe_img21\"<\/a><\/p>\n

Answer = (D)<\/strong>.<\/p>\n

7) This one becomes clearer if we change the roots to fractional exponents.<\/p>\n

\"cgpwe_img22\"<\/a><\/p>\n

Answer = (A)<\/strong>.<\/p>\n

8) The exponent of 3, which is 42, is close to 40.\u00a0 If 2 and 3 had exponents of 60 and 40, respectively, those would be in a ratio easy to reduce — 60:40 = 3:2.\u00a0 Clearly<\/p>\n

\"cgpwe_img23\"<\/a><\/p>\n

Raise both sides of that inequality to the power of 20.<\/p>\n

\"cgpwe_img24\"<\/a><\/p>\n

Therefore, III is bigger than I.<\/p>\n

Now, let’s think about II.\u00a0 The square root of 2 is 2 to the power of 1\/2, so 2 times the square root of two, the contents of the parentheses, would be 2 to the power of 1.5.\u00a0 Multiply the exponents: 1.5*35 = 35 + 17.5 = 52.5 — that would be the resultant exponent of 2.\u00a0 Clearly, this is a lower power of 2 than given in statement I.\u00a0 So, II is less than I.<\/p>\n

From smallest to biggest is II, I, III.<\/p>\n

Answer = (C)<\/strong>.<\/p>\n

9) An increase of 50% corresponds to a multiplier<\/a> of 1.5, so the above information can be written as<\/p>\n

\"cgpwe_img25\"<\/a><\/p>\n

To get all the K’s on one side, we will divide by the power on the left, K to the power of 5\/4.\u00a0 This will mean we have to subtract the exponents.\u00a0 What do we get when we subtract 5\/4 from 3\/2?<\/p>\n

\"cgpwe_img26\"<\/a><\/p>\n

Thus, after the division, we have:<\/p>\n

\"cgpwe_img27\"<\/a><\/p>\n

Now, raise both sides to the fourth power to solve for K:<\/p>\n

\"cgpwe_img28\"<\/a><\/p>\n

Answer = (E)<\/strong>.<\/p>\n

10) First, we have to express 0.15 as a fraction:<\/p>\n

\"cgpwe_img29\"<\/a><\/p>\n

Also, 16 is 2 to the power of 4, so the equation with b tells us:<\/p>\n

\"cgpwe_img30\"<\/a><\/p>\n

Bases are the same, so we can equate the exponents.<\/p>\n

\"cgpwe_img31\"<\/a><\/p>\n

Answer = (E)<\/strong>.<\/p>\n

11) As is usual, all the answer choices have been rationalized, so we have to rationalize the denominator<\/a> of the prompt fraction.\u00a0\u00a0 This means multiplying by the conjugate over itself: that would be 7 minus<\/em> 3 times the square root of 5.<\/p>\n

\"cgpwe_img32\"<\/a><\/p>\n

In the denominator, we use the difference of two squares<\/a> pattern.\u00a0\u00a0 In the numerator, we simply FOIL<\/a>.<\/p>\n

\"cgpwe_img33\"<\/a><\/p>\n

Answer = (D)<\/strong>.<\/p>\n

BTW, you don’t need to know this for the GMAT, but this OA answer is the reciprocal of the Golden Ratio<\/a>, and answer choice (E)<\/strong> equals the Golden Ratio.<\/p>\n

12) We need to express each side as a power of 5.\u00a0 We will use fractional exponents for the roots.<\/p>\n

\"cgpwe_img34\"<\/a><\/p>\n

Equate the exponents.<\/p>\n

\"cgpwe_img35\"<\/a><\/p>\n

Multiply both sides by 6 to clear the fractions.<\/p>\n

\"cgpwe_img36\"<\/a><\/p>\n

Answer = (B)<\/strong>.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

This post was updated in 2024 for the new GMAT. Here are twelve challenging problems related to the topic of exponents & roots.\u00a0 Remember, no calculator. (A) 17 (B) 19 (C) 21 (D) 23 (E) 27 4) Rank the following quantities in order, from smallest to biggest. (A) I, II, III (B) I, III, II […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[112],"tags":[],"ppma_author":[13209],"class_list":["post-5198","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"yoast_head":"\nChallenging GMAT Problems with Exponents and Roots - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Challenging GMAT Problems with Exponents and Roots\" \/>\n<meta property=\"og:description\" content=\"This post was updated in 2024 for the new GMAT. Here are twelve challenging problems related to the topic of exponents & roots.\u00a0 Remember, no calculator. (A) 17 (B) 19 (C) 21 (D) 23 (E) 27 4) Rank the following quantities in order, from smallest to biggest. (A) I, II, III (B) I, III, II […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2014-11-03T17:00:54+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-01-15T18:48:28+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gmat\/files\/2014\/10\/cgpwe_img1.png\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"12 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"Challenging GMAT Problems with Exponents and Roots\",\"datePublished\":\"2014-11-03T17:00:54+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\"},\"wordCount\":944,\"commentCount\":13,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/challenging-gmat-problems-with-exponents-and-roots\/\",\"name\":\"Challenging GMAT Problems with Exponents and Roots - 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":13209,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/5198","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/comments?post=5198"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/posts\/5198\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/media?parent=5198"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/categories?post=5198"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/tags?post=5198"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/gmat\/wp-json\/wp\/v2\/ppma_author?post=5198"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}